Title Hunt, J.; Belcher, S.; Stretch, D.; Sajjadi, S.; Clegg, J.;...

Title Gas Transfer at Water Surfaces 2010( p.38 )

Author(s)

Hunt, J.; Belcher, S.; Stretch, D.; Sajjadi, S.; Clegg, J.;Kitaigorodskii, S.A.; Toba, Y.; Turney, D.; Banerjee, S.;Janzen, J.G.; Schulz, H.E.; Gonzalez, B.C.G.; Lamon, A.W.;Hughes, C.; Drennan, W.M.; Kiefhaber, D.; Balschbach, G.;Abe, A.; Heinlein, A.; Lee, G.A.; Jirka, G.H.; Kurose, R.;Waite, C.; Onesemo, P.; Ninaus, G.; Choi, Y.J.; Takahashi, K.;Baba, Y.; Komori, S.; Ohtsubo, S.; Takagaki, N.; Iwano, K.;Handa, K.; Shimada, S.; Akiya, Y.; Beya, J.; Peirson, W.;Banner, M.; Nezu, I.; Mizuno, S.; Sanjou, M.; Garbe, C.S.;Toda, A.; Takehara, K.; Takano, Y.; Etoh, T.G.; Caulliez, G.;Hung, L.-P.; Tsai, W.-T.; Tejada-Martínez, A.E.; Akan, C.;Khatiwala, S.; Grosch, C.E.; Jayathilake, P.G.; Khoo, B.C.;Rocholz, R.; Tan, Z.; Nicholson, D.P.; Leifer, I.; Emerson,S.R.; Hamme, R.C.; Mischler, W.; Simões, A.L.A.; Jähne, B.;Patro, R.; Loh, K.; Cheong, K.B.; Uittenbogaard, R.; Jeong, D.;Mori, N.; Nakagawa, S.; Soloviev, A.; Fujimura, A.; Gilman,M.; Hühnerfuss, H.; Monahan, E.C.; Haus, B.; Savelyev, I.;Matt, S.; Donelan, M.; Rhee, S.H.; Vlahos, P.; Huebert, B.J.;Edson, J.B.; Richter, K.E.; McNeil, C.L.; Yan, X.; Walker,J.W.; Zappa, C.J.; Ribas-Ribas, M.; McGillis, W.R.; Schimpf,U.; Rutgersson, A.; Nagel, L.; Orton, P.M.; D'Asaro, E.A.;Nystuen, J.A.; Gómez-Parra, A.; Forja, J.M.; Smedman, A.-S.;Sahlée, E.; Pettersson, H.; Kahma, K.K.; Perttilä, M.; Park, G.-H.; Chelton, D. B.; Bell, T.G.; Risien, C.M.; Kondo, F.;Suzuki, N.; Suzuki, Y.; Wanninkhof, R.; Johnson, M.T.;Campos, J.R.; Liss, P.S.; Tsukamoto, O.; Wanner, S.

Citation Kyoto University Press. (2011)

Issue Date 2011-07-04

URL http://hdl.handle.net/2433/156156

Right Copyright (C) S. Komori, W. McGillis, R. Kurose, KyotoUniversity Press 2011

Type Book

Textversion publisher

Kyoto University

Marvellous self-consistency inherent in wind waves

— Its origin and some items related to air–sea transfers

Yoshiaki Toba

Tohoku University (Emeritus); Home: 228-91 Iwakura-Nakamachi, Sakyo-ku,

Kyoto, 606-0025 Japan, E-mail: [email protected]

Abstract. There exists conspicuous self-consistency inherent in wind waves

which are in local equilibrium with the wind, or in the wind‒windsea

equilibrium. From my long-term concern, I will review such aspects based on my

present ideas. Aspects of the self-consistency are: (1) the 3/2-power law for

significant waves, (2) the wave spectral form proportional to the friction velocity

and the frequency to the minus fourth power, (3) proportionality of the Stokes

drift with the friction velocity, and (4) the existence of the downward bursting

boundary layer (DBBL), where turbulent intensities are proportional to the

friction velocity. These aspects have very simple forms, and are mutually

convertible, to form a holistic nature. The origin of these characteristics is

discussed with the concept of the wave breaking adjustment. In relation to the

air‒sea transfers of momentum and gas, importance of the windsea Reynolds

number is stated, and direction of further researches is recommended.

Key words: 3/2-power law, wave spectrum, Stokes drift, friction velocity,

DBBL, wave breaking adjustment, windsea Reynolds number

1. Aspects of self-consistency in wind waves

1.1 The 3/2-power law

The 3/2-power law is expressed as:

H=BT3/2, B=0.062, (1)

where H=gH s/u2, and T=gT s/u, with Hs, the significant wave height, Ts, the

significant wave period, u the air friction velocity, and B is a constant. (1) was

found (Toba 1972), by eliminating nondimensional fetch from empirical fetch

graph formulas by Wilson (1965), Mitsuyasu et al. (1971) and Hasselmann et al.

(1973), together with dimensional considerations.

Equation (1) may alternatively be expressed by a dimensional form:

H s=B(gu)12T s

3/2, B=0.062, (2)

38

or, by using total energy E and the spectral peak angular frequency σp,

E=Bσ3p , B=0.051, (3)

where E=g2E/u4, σ

p =uσp/g (Toba 1978). Also, Bailey, Jones and Toba

(1991) converted (1) to a relation between significant wave steepness and the wave

age.

In the actual sea, variability of the wind, and more or less swell exist. In the

observation data, the above coefficient B in (1) or (2) actually has deviation within

20% or so (Toba et al 1990). Also, Ebuchi et al. (1992) investigated the effect of

swells and changing winds, by using data from ocean data buoy stations. This

variation of B may also be caused in the estimation of u, since it includes the sea

surface drag coefficient CD in many cases in course of the estimation.

As seen in Figure 1, this law (1) was also applicable to laboratory data by Toba

Marvellous self-consistency inherent in wind waves ̶ Its origin and some items related to air‒sea transfers 39

102

101

100

10-1

H* = 0.062T*3/2

Laboratory (Toda, 1961)Tower Stn. (Kawai et al., 1977)do., including swell

100 101 102

T *

H*

Converted from formulas by Wilson (1965) Mitsuyasu et al. (1971) Hasselmann et al. (1973)

Figure 1 The 3/2-power law for windsea in local equilibrium

with the wind in the form of (1), cited from Toba, Suzuki

and Komori (2008). The data was originally taken from

Kawai, Okada and Toba (1977), and the spurious triangle

for u and a triangle for the slope of data-point distribution

are added. Closed symbols indicate observation data, and

open symbols are used to show three continuous lines,

which are very close to one another, converted from three

empirical formulas.

(1961) and to tower station data compiled by Kawai et al. (1977). More recently,

e.g., Lamont-Smith and Waseda (2008) reported, by using observation data, that

this law is satisfied effectively, throughout from a wind wave tunnel and to the

ocean, though the wave height and the frequency develop by their respective laws.

The empirical formulas of the wind sea growth by Wilson (1965) had complex

forms as functions of the nondimensional fetch, where wind sea approaches the

fully developed state. Nevertheless such a complicated form was converted to be a

straight line as shown in Figure 1. This led us to a concept of the wind sea in local

equilibrium with the wind or the wind‒windsea equilibrium, as described in

Section 2.

1.2 The guσ−4 -type power spectrum

The high frequency side of self-similarity one-dimensional spectral form of

wind waves is expressed by the following form, including u-proportionality:

φ(σ)=αsguσ4, σ>σp, αs=0.062±0.010 (4)

where αs is a constant, σp the peak angular frequency, g=g (1 + Sκ2/ρw) is the

acceleration of gravity extended to include the effect of surface tension S for high

frequency part, with κ the wave number and ρw is the water density (Toba 1973).

This spectral form was derived in order for the integral of (4) to correspond to (1),

and at the same time in accordance with observations by Toba (1961) and Kawai et

al. (1977).

Kitaigorodskii (1983) and Phillips (1985) re-proposed this spectral form

theoretically, and (4) was widely accepted by many observations by Mitsuyasu et

al. (1980), Kahma (1981), and others, as reviewed by Janssen (2004). Phillips

(1985) named αs Toba’s constant, and stated that the observed values were ranged

from 0.06 to 0.11. Janssen (2004) reported a value of 0.127 based on many

observations. More recently Romero and Melville (2010) have presented

observational data showing an increasing trend from 0. 06 to 0. 11 with the

increasing wave age.

It should be noted, however, that as was clearly shown in Toba, Okada and

Jones (1988), αs has fluctuating values caused by unsteadiness of the wind,

showing undersaturation and oversaturation of the spectral level, since it takes

some time for wind waves to reach the wind‒windsea equilibrium to the new

wind. Consequently, αs varies inversely to the fluctuation of u. Also, for

decreasing wind speed, the spectral peak shifts to a little lower frequency side with

higher values. For increasing wind, it shifts to a little higher frequency side to have

a little lower value, as we reported in the same paper. Hanson and Phillips (1999)

supported this by new observation data.

Zakharov and Filonenko (1966) had predicted theoretically that the one-

Section 1: Interfacial Turbulence and Air‒Water Scalar Transfer40

dimensional spectral form of wind waves should be proportional to σ−4, for surface

gravity wave interactions, and (4) is in accordance with this theory. However,

present day importance of the spectral form (4), which includes the u -

proportionality, is that it has become the basis of satellite observations of the sea

surface wind (see, e.g., Kawamura, 2003).

1.3 Stokes drift–friction velocity proportionality

The Stokes drift of wind waves as the water waves, u0, is expressed by

u0=π3H s

2/gT s3 . (5)

Combining (5) with (2), we get immediately its u-proportionality (Toba 1972):

u0=π 3B2u=0.12u . (6)

This u-proportionality should be applicable from laboratory to ocean waves, and

by the self-similar spectral form of (4), for all the stages of developing wind

waves, and also to the individual waves in a sense shown by Tokuda and Toba

(1981).

Bye (1988) also reported that u0 is proportional to u for the spectral form (4),

and predicted theoretically with observation that the wind drift surface current us is

approximately twice u0.

1.4 Friction velocity–turbulence intensity proportionality within DBBL

Turbulence characteristics also have proportionalities with u. From our

laboratory experiment data (Yoshikawa et al. 1988), Toba and Kawamura (1995)

found a special boundary layer below the wind waves: the downward-bursting

boundary layer (DBBL), where (7) holds:

(u 'a2)

12∝(u 'w

2)12∝uw∝u∝u0 , (7)

where (u 'a2)

12is the intensity of air turbulence, (u 'w

2)12

that of water turbulence,

and uw is the water friction velocity. DBBL has observationally a thickness of

about 5 times Hs, and in DBBL the u-proportionality of turbulence intensities and

Reynolds stress holds.

Figure 2 shows, cited from Toba and Kawamura (1995), the depth distribution

in z/Hs, of turbulence intensities (a) and Reynolds stress (b) in water, showing the

existence of this DBBL. In (b), the value of 0.33 of the abscissa, shown by the

vertical line, almost exactly corresponds to (6), since uw/u0=0.33 of the Figure

correspond to u0/u=0.12.

Figure 3 shows, cited also from the same paper, four data sets of the depth of

the DBBL. Five thin segments of the Figure were taken from Toba et al. (1975)


of a wind-wave tank, where DBBL developed as soon as wind waves were

generated, in spite that no air-entrainment occurred yet, as seen from their Photos 1

to 3. A rectangle in the middle part is from an experiment of air‒water CO2

exchange in a wind wave tank by Komori et al. (1995). The three broken segments

show observations of bubble clouds in the sea cited from Thorpe (1986; 1992).

Kitaigorodskii et al. (1983), using Donelan’s data in Lake Ontario, reported


10.0

0.1

1.0

10.0

100.00.0 1.0 0.0 1.0

(a) (b)

z/H

s

10.0

0.1

1.0

10.0

100.0

z/H

s

u´ w´U1U2U3

U1U2U3

( u´2, w´2)/u* -u´w´/u*

10

5

00 1 2 3 4 2 3 4 5

Hs (cm) Hs (m)

D/H

s

Figure 2 The depth distribution, in z/Hs, of turbulence intensities (a) and Reynolds stress

(b) in water, showing the existence of this DBBL. Cited from Toba and Kawamura

(1995), which reanalyzed Yoshikawa et al. (1988) data.

Figure 3 Four data sets of the depth of the DBBL, D, normalized by Hs, cited from

Toba and Kawamura (1995). The data set includes, from the left to the right, a

laboratory no bubble entrainment case by Toba et al. (1975), the turbulence

measurement data by Yoshikawa et al. (1988), the air‒water CO2 exchange data by

Komori et al. (1995) in a laboratory, and observations of bubble clouds in the sea

cited from Thorpe (1986; 1992).

the two-layer structure, with the upper layer of the order of ten times the rms. wave

amplitude, having intense turbulence generation by wind waves. This should just

correspond to the DBBL. More recently, Yoshioka et al. (2003) made detailed

observations of wind sea, breaking waves, swell, together with bubble entrainment

depth, and reported a similar structure with our DBBL.

2. Origin of the self-consistent aspects – Wave-breaking adjustment

The above described four aspects of wind sea are all in very simple forms, and

yet they are mutually consistent or convertible. Thus the wind sea seems to have a

holistic structure. Now what is the origin of this marvellous self-consistent nature.

Toba (1988; 1996) gave some comprehensive discussion at that time, and Csanady

(1997), Badulin et al. (2005; 2007), Kukulka and Hara (2008a; 2008b), for

example, seem to have pursued their own approaches.

From my present time idea, the simplest form among the above characteristics

will be (6) and (7). The water surface is driven by the wind stress, represented here

by u, and the water surface elements of individual waves must always adjust

themselves to move with the same speed u0. This should be performed by the wave

breaking, whether or not bubble entrainment is taking place. Here u0 should be

taken relative to the water layer below the DBBL.

In the wind‒windsea equilibrium or in wind waves in local equilibrium with

the wind, waves and turbulence are coupled by strongly nonlinear processes,

including the skin friction distribution along the phase of waves (Okuda et al.

1977; Banner and Peierson, 1998), local shear flows caused by it, wave breaking

(with or without bubble entrainment), ordered motions in the air and water

(Kawamura and Toba 1988), bursting and turbulence (Ebuchi et al. 1993). If the

velocity of some wave element exceeds the above speed u0, it breaks to satisfy this

requirement. This situation may be expressed as the wave-breaking adjustment.

The above characteristics of wind waves were also derived in a quite different

manner (Toba 1974). Dimensionally obtained Kolmogoroff and Obukhov’s law in

turbulence was expressed by

V 3/Λ=constant, (8)

where V is the velocity difference over a distance Λ of the size of an eddy. When

this was applied to wind waves as Λ: 2a=H, V: 2u=2σa=Hσ, where a is the

wave amplitude, and using a dimensional supposition of H2σ3= u g, we

immediately arrive at

H=(T/2π)32

. (9)


This is equivalent to the 3/2-power law with

B=(2π)32

=0.0635(≈0.062) and u0=π3B2u=(2)

3u=0.125u . (10)

Wind waves thus seem to have duality of wave and turbulence.

It should be noted that Tulin (1996) suggested that the 3/2-power law can be

explained by a simultaneous consideration of wave energy and wave momentum

change rates resulting from wave breaking, keeping the water-wave relation:

M=E / C, where M is the wave momentum, E the wave energy, and C is the phase

speed. This leads to the necessary downshifting of wave frequency, to satisfy the

3/2-power law, keeping a balance between the wind input and dissipation by wave

breaking.

Observationally, the part of momentum, that is retained as the wave

momentum, is 6% in short fetch wind-wave tanks, and it decreases with the wave

age by an error functional way down to zero value for the fully developed sea

(Toba 1978). However, a purely theoretical approach from these elementary

processes, including the wind, wave breaking and turbulence, may not be easy,

because of their strongly nonlinear holistic nature. Elucidation of how the stress or

works by the wind remains in waves and goes through the DBBL to the lower

layer, would be future works.

3. Windsea Reynolds number RB and air–sea transfers

By the above story of consistency, H, T, or the wave age, σp=uσp/g, are

unified one nondimensional quantity describing the state of wind sea for given u.

There is the other quantity that should describe the fluid dynamical state of wind-

sea surfaces. It is the windsea Reynolds number:

RB=u2/νσp , (11)


Figure 4 Schematic picture of the microphysical processes at the air‒water

interface. (Cited from Toba 1996).

wind

high shear layer

air flowseparation

high vorticityregion

ordered motion in waterburst

airentrainment

reattachment

ordered motion in air

where ν is the kinematic viscosity of air.

The RB was originally introduced by Toba and Koga (1986) as a

nondimensional parameter describing the overall conditions of air‒sea boundary

processes, including the percentage of the breaking wave crests, the whitecap

coverage, and the concentration of sea-salt particles on the sea surface. Iida, Toba

and Chaen (1992) presented further data showing that the production rate of sea

water droplets on the sea surface can be expressed well by using this. Toba, Suzuki

and Iida (1999) used RB in the estimates of global distribution of whitecap

coverage and sea-salt production on the sea surface, using the terminology of the

breaking wave parameter for RB.

Synthesizing these studies, Figure 5 clearly shows that the percentage α of

breaking crests among individual waves of windsea passing a fixed point on the

water surface are widely distributed when plotted against U10, and it collapses


(a) (b)

(c) (d)

50

40

30

20

10

00 5 10 15 20 25

U (ms-1)

α (%

)

α (%

)

FETCH = 13.6m

10.0 Toba (1972)

6.9

TOWER ST. — Toba et al. (1971)

FETCH = 13.6m

10.0 Toba (1972)

6.9

Toba et al. (1971)

Koga (1981)

class 5 (2.75 < log m < 3.25)

class 6 (3.25 < log m < 3.75)

class 7 (3.75 < log m < 4.25)

class 8 (4.25 < log m)

102

101

100

103 104

RB

101

100

10-1

10-2

10-3

10-4

101 102 103 104 105 106 102 103 104 105

RB RB

Toba (1972)

Toba et al. (1971)

Ross and Cardone (1974)

Toba and Ckaen (1973)

Snyder et al. (1983)

Monahan (1971)

105

104

103

102

θ̂ c (

cm-3

)

W (

%)

Figure 5 Cited from Toba et.al. (2006), (a) Percentage α of breaking crests among individual

waves of wind sea passing a fixed point on the water surface, plotted against U10, and (b) plotted

against RB (from Toba and Koga 1986). (c) Whitecap coverage percent, W, plotted against RB

(from Zhao and Toba 2001). (d) Sea-water droplet concentration observed near the sea surface,

plotted against RB (from Iida et al. 1992).

when plotted against RB. At the same time it is seen that there is a clear critical

value of RB at 103, where breaking wave and drop production begin to take place.

Further, Zhao and Toba (2001) showed the clear advantage of RB over U10 , u and

the wave age, in describing observed data sets of whitecap coverage. Toba et al.

(2006) presented synthesizing review of the derivation and physical interpretation

of RB, and gave the new terminology of the windsea Reynolds number. As a matter

of fact, RB may be interpreted as a Reynolds number, since it contains u as a

representative scale of speed, and Ls=uTs (=2πu/σp) as a representative length

scale of the phenomenon under consideration, and ν.


10-2

10-3

100 101 102 103 104 105 106

RB

CD

Hamada (1963)Kunishi (1963)Kunishi & Imasato (1966)Toba (1961, 1972)Masuda & Kusaba (1987)Banner and Peirson (1998)Komori et al. (1999)

Figure 6 Data set of the drag coefficient CD, measured by wind-wave tank

experiments in fetch-limited conditions, plotted against the windsea

Reynolds number RB. (Cited from Toba et al. 2006.)

smooth surface (small RB)

transitional (2×102 < RB < 103)

rough surface (RB103)

breaking wave (RB > 103)

skin friction

form drag

Figure 7 Schematic representation of the regime shift of the

windsea boundary layer conditions as a function of RB, cited

from Toba et al. (2006).

This is an appropriate quantity also to study and describe air‒water momentum

or gas transfers. Zhao et al. (2003) effectively expressed the CO2 transfer velocity

as a function of RB, and Suzuki et al. (2001) attempted a global mapping of the

air‒sea CO2 exchange by using this kind of formula. Zhao et al. (2006) presented a

now sea-spray generation function for spume droplets as a function of RB. Soloviev

et al. (2007) reported further researches using RB.

Figure 6 shows measured values of CD in laboratory tanks by many authors

(cited from Toba et al. 2006). We can see two critical points in the distribution of

CD, at RB=2×102 and at RB=103, where the data indicates two downward points,

which bracket a bulge region. Beyond RB=103, CD values become large. The latter

point corresponds well to the critical point of wave breaking shown in Figure 5.

Figure 7 shows, from Toba et al. (2006), schematic representation of the

regime shift of the windsea boundary layer conditions as a function of RB. The

transitional regime between 2×102 and 103 in RB corresponds to the regime where

flow separation is large with the form drag leading over the skin friction, in

accordance with data by Banner and Peirson (1998).

When we add data observed in the sea where some swells usually exist, point

distribution in Figure 6 becomes less sharp, as Toba et al. (2006) had shown.

Nevertheless CD vs. RB expression seems much better than, e.g., the

nondimensional roughness parameter v.s. wave age expression proposed in Jones

and Toba (Eds.) (2001) book. Suzuki et al. (2010) has further investigated the

effect of swell, using observation tower data including two-dimensional wave

spectra. In this sense, well-designated studies of gas transfer as well as momentum

transfer should be necessary in the future, especially for cases where various

swells coexist.

4. Concluding remarks

Throughout the whole stages of wind sea development, the strokes drift of

individual waves adjust themselves to go with the same velocity, u0=π3B2u in

the mass, performed by the wave-breaking adjustment. This results in the 3/2-

power law, the guσ−4-type power spectrum, and the DBBL below wind sea

having a special turbulence structure. These altogether constitute the self

consistent, holistic nature of the wind sea.

The windsea Reynolds number RB acts as a good parameter to describe the

hydrodynamic regime of the surface of the wind sea, and can serve to describe

characteristics of variation of air‒water transfer coefficients of momentum and

gasses. However, where there are prevailing swells, they affect largely the

exchange coefficients. Therefore we should accumulate more data of air‒sea

fluxes by further intentional experiments and observations, in which two-


dimensional wave spectra are included.

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